opimization of space trajectories: invariant manifolds +...
TRANSCRIPT
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Optimization of Space Trajectories:
Invariant Manifolds + DMOC
Ashley MooreJuly 8, 2008
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Outline
Introduction DMOC
Overview Motivating Example
DMOC + Invariant Manifolds Invariant Manifolds
Basic idea and details “Shoot the Moon”
Bicircular Model of Four Body Problem Overview Equations of motion Results
DMOC details for “Shoot the Moon” DMOC Results Future Work References
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Introduction
Objective: Design a low energy space trajectory Use Invariant Manifold techniques to determine
initial trajectory Apply DMOC to generate an optimal solution
“Shoot the Moon” Test method by designing trajectory from Earth
to Moon Split problem into two coupled planar circular restricted
3-body systems and patch them together Sun - Earth - Spacecraft (SE) Earth - Moon - Spacecraft (EM)
Based on PhD thesis of Shane Ross and “Shoot the Moon” paper by Koon, Lo, Marsden, and Ross
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DMOC Overview
DMOC is based on a direct discretization of the Lagrange-d’Alembert principle for a dynamical system Produces the forced discrete Euler-Lagrange
equations Serve as optimization constraints given a cost
function Need good initial guess that obeys dynamics to work
successfully
Junge, O., Marsden, J.E., and Ober-Blöbaum. “Discrete Mechanics and Optimal Control.”
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DMOC Motivating Example
Orbit Problem Goal: Optimally move a spacecraft from circular orbit r = 5
to r = 10 with 2 revolutions around the earth. Minimize the control effort
Lagrangian
Force
Cost function€
L q, ˙ q ( ) = 12
m ˙ r 2 + r2 ˙ ϕ 2( ) + GMmr
€
f =0ru
€
J q,u( ) = u t( )2dt0
T∫ Junge, O., Marsden, J.E., and Ober-
Blöbaum. “Discrete Mechanics and Optimal Control.”
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DMOC
Optimal Trajectory
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DMOC Motivating Example
What if the desired trajectory looks like this:
DMOC will need an excellent initial guess
Earth
Moon’s Orbit
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DMOC + Invariant Manifolds
Invariant Manifold method generates initial condition (patch point) Integrate patch point in Bicircular 4 body model for initial trajectory Apply initial trajectory to DMOC using same model What should be minimized?
Depends on payload If people - minimize time or distance If supplies/robotics - minimize fuel
Constraints Euler-Lagrange equations Initial position and momentum Final position and momentum
What do we expect? Perhaps DMOC will generate trajectory with gradual ΔV instead of
concentrated ΔV at patch point Shorter flight time or distance
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Invariant ManifoldsBasic Idea
Stable and unstable manifolds emanate from the periodic orbits of Lagrange points of the PCR3BP
Manifold tubes connect regions of space Spacecraft may travel from one region to another
through tubes
Ross, S.D., “Cylindrical Manifolds and Tube Dynamics in the Restricted Three-Body Problem” (PhD Thesis, California Institute of Technology, 2004), pp. 121.
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Invariant ManifoldsDetails
Use rotating coordinate system centered on barycenter of m1 and m2.
Normalize system using mass parameter
Neglect spacecraft mass PCR3BP equations
€
µ =m2
m1 + m2 where m1 > m2
€
˙ ̇ x − 2 ˙ y =Ωx˙ ̇ y + 2 ˙ x =Ωy
€
Ω =x 2 + y 2
2+1−µr1
+µr2
Ross, S.D., “Cylindrical Manifolds and Tube Dynamics in the Restricted Three-Body Problem” (PhD Thesis, California Institute of Technology, 2004), pp. 8.
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Invariant ManifoldsDetails
€
U x,y( ) = − 12
µ1r12 + µ2r2
2( ) − µ1r1−
µ2r2
µ1 =1−µ, µ2 = µ €
E x, y, ˙ x , ˙ y ( ) = 12
˙ x 2 + ˙ y 2( ) + U x,y( )
Hill’s Regions
Ross, S.D., “Cylindrical Manifolds and Tube Dynamics in the Restricted Three-Body Problem” (PhD Thesis, California Institute of Technology, 2004), pp. 14.
Energy Integral
Energy divides the phase space into regions The energy restricts the motion
of a spacecraft
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Invariant Manifolds“Shoot the Moon”
Locate L2 Lagrange point for the SE and EM systems Compute periodic orbit and ‘grow’ manifolds
Sun-Earth Manifolds Earth-Moon Stable Manifold
Moon
Earth
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Invariant Manifolds“Shoot the Moon”
Transform EM manifold into SE rotating coordinates and plot manifolds together
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Invariant Manifolds“Shoot the Moon”
Compute Poincaré Sections and select ‘patch’ point Select point just outside Sun-Earth manifold and inside
Earth-Moon manifold
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Invariant Manifolds“Shoot the Moon”
Use selected point as initial condition Integrate forwards on Earth-Moon stable manifold Integrate backwards on Sun-Earth unstable manifold
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Invariant Manifolds“Shoot the Moon”
Capture at Moon occurs naturallyEM Trajectory in EM Rotating Coordinates
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Bicircular Model
M1 and M2 rotate in circular motion about their barycenter
M0 and M1-M2 barycenter rotate in circular motion about their common center of mass
Create similar trajectory using the Bicircular Model of the four body problem (BCM4)
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Bicircular Model
Sun Earth Rotating system:
€
˙ x = u˙ y = v
˙ u = x + 2v −µE x − xE( )
x − xE( )2
+ y 2( )3
2−
µS x − xS( )
x − xS( )2
+ y 2( )3
2−
µM x − xM( )
x − xM( )2
+ y − yM( )2( )
32
˙ v = y + 2u − µE y
x − xE( )2
+ y 2( )3
2−
µS y
x − xS( )2
+ y 2( )3
2−
µM y − yM( )
x − xM( )2
+ y − yM( )2( )
32
€
µ =ME
ME + MS= 3.0035 ×10−6
µS =1−µ µE = −µ
µM = 3.734 ×10−8
xS = −µ xE =1−µ
€
aM = 2.573×10−3
ωM =12.369θM =ωM t + θM 0xM = aM cos θM( )yM = aM sin θM( )
www.esm.vt.edu/~sdross/books
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Bicircular Model
Trajectory Start at 800 km
circular Earth orbit ΔV =175.8 m/s
Initial Guess: Trajectory
Initial Guess: Control Force
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Trajectory Sensitivity
ΔV = 207 m/s ΔV = 196 m/s ΔV = 193 m/s
ΔV = 192.8 m/s ΔV = 191 m/s ΔV = 188 m/s
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DMOC+IM
Lagrangian is derived from BCM4 in SE rotating coordinates
DMOC equations
Minimize control effort
Control Force
€
L = 12
˙ x 2 + ˙ y 2( ) + 12 x2 + y 2( ) + x˙ y − y˙ x + µE
x − xE( )2
+ y 2+
µM
x − xM( )2
+ y − yM( )2
+µS
x − xS( )2
+ y 2
€
q0 = q0 qN = q
1
D2L q0, ˙ q 0( ) + D1Ld q0,q1( ) + f0− = 0D2Ld qk−1,qk( ) + D1Ld qk,qk +1( ) + fk−1+ + fk− = 0 for k =1,...,N −1 −D2L qN , ˙ q N( ) + D2Ld qN−1, ˙ q N( ) + fN−1+ = 0
€
ux =ΔVxΔt
, uy =ΔVyΔt€
J(q,u) = ux t( )2
+ uy t( )2dt
0
T∫
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DMOC Results
Trajectory Control Force
IG 175.8273DMOC 1 2.1374DMOC 2 0.6105DMOC 3 0.2342DMOC 4 0.2331
DeltaV (m/s)
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DMOC Results
Initial Guess DMOCcase 1 175.8273 0.2331case 2 178.5763 0.4452case 3 172.7951 0.0672case 4 171.3516 0.0902case 5 177.8498 0.4386
Delta V (m/s)
Initial Guess DMOC Result
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Comparison
How does this compare with a Hohmann Transfer? Case 1: trajectory begins in ~800 km altitude circular
orbit. Starting velocity of trajectory = 6.24 km/s circular velocity of parking orbit = 7.4 km/s
Initial ΔV = 1.17 km/s ΔV = 0.2331 m/s for trajectory portion Total ΔV = 1170.23 m/s
Hohmann Transfer from 800 km circular orbit to Moon Total ΔV = 3812.6 m/s
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DMOC + Invariant ManifoldsFuture Work
Optimize for time and control Enforce momentum boundary
conditions to ensure capture Solve same problem using JPL’s
MYSTIC compare with DMOC+IM method
Use method to generate trajectory to Titan Also include fly-by of Enceladus
May require additional maneuvers
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References
Gomez, G., Koon, W.S., Lo, M.W., Marsden, J.E., Masdemont, J., and Ross, S.D. “Invariant Manifolds, the Spatial Three-Body Problem and Space Mission Design.” AAS 01-301.
Junge, O., Marsden, J.E., and Ober-Blöbaum, S. “Discrete
Mechanics and Optimal Control.”
Koon, W.S., Lo, M.W., Marsden, J.E., and Ross, S.D. “Dynamical
Systems, the Three-Body Problem and Space Mission Design.”
www.esm.vt.edu/~sdross/books
Koon, W.S., Lo, M.W., Marsden, J.E., and Ross, S.D. “Shoot the
Moon.” AAS 00-166.
Koon, W.S., Lo, M.W., Marsden, J.E., and Ross, S.D.
“Heteroclinic Connections Between Periodic Orbits and
Resonance Transitions In Celestial Mechanics.” Chaos 10
(2000): 427-469.
Ross, S.D., Koon, W.S., Lo, M.W., and Marsden, J.E. “Design of a Multi-Moon Orbiter.” AAS 03-143.
Ross, S.D. “Cylindrical Manifolds and Tube Dynamics in the
Restricted Three-Body Problem.” PhD thesis, California
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Questions?